One approach to modeling beams in the source plane is to create
mock observations with point sources, reconstruct those sources,
and fit Gaussians to measure beam properties [1]}. As an alternative that does not require extensive reconstructions
with mock data, we can identify a matrix that describes the
source-plane resolution elements. Consider starting with some
source \(\mathbf {s}_0\) , creating a mock observation with noise, and then
performing a reconstruction. The mock data can be written as
\(\mathbf {d}= \mathbf {L}\mathbf {s}_0 + \mathbf {n}\)  where \(\mathbf {n}\)  is a vector of
noise in the data. Using Equations REF  and REF ,
we can write the reconstructed (most probable) source as
\(\mathbf {s}_{\text{MP}} &= \mathbf {A}^{-1} \mathbf {L}^\top \mathbf {C}_\textrm {d}^{-1} \mathbf {d}\nonumber \\&= \mathbf {A}^{-1} \mathbf {L}^\top \mathbf {C}_\textrm {d}^{-1} \mathbf {L}\mathbf {s}_0 + \mathbf {A}^{-1} \mathbf {L}^\top \mathbf {C}_\textrm {d}^{-1} \mathbf {n}\nonumber \\&= \mathbf {A}^{-1} \mathbf {F}\mathbf {s}_0 + \mathbf {A}^{-1} \mathbf {L}^\top \mathbf {C}_\textrm {d}^{-1} \mathbf {n}.\) 
